Universität Konstanz Sommersemester 2013 Fachbereich Mathematik und Statistik

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Universität Konstanz Sommersemester 2013 Fachbereich Mathematik und Statistik

Prof. Dr. Stefan Volkwein

Roberta Mancini, Stefan Trenz, Carmen Gräßle, Marco Menner, Kevin Sieg

Optimierung

http://www.math.uni-konstanz.de/numerik/personen/volkwein/teaching/

Sheet 5

Deadline for hand-in: 2013/06/24 at lecture Exercise 13

Let f, g : R → R be four-times differentiable functions where g is an approximation of f : g(x) = f(x) + r(x)

with some error function r : R → R and

|r(x)| ≤ for some known > 0.

Determine the numerical first derivative D

_c¹

(g, h) of g by central differences, where h denotes the grid spacing. Compute the error arising in the numerical approximation of the first derivative. Use this result to show that the error arising in the numerical approximation of the second derivative by using again central differences is of order O(

⁴⁹

).

Exercise 14

Let f ∈ C

²

( R

ⁿ

, R ) and consider a uniform n-dimensional mesh with spacing h and e

_i

∈ R

ⁿ

, i = 1, ..., n, the canonical ith basis vector. Verify the (2nd order forward difference) formula

∂

∂x

_i

∂

∂x

_j

f(x) = f (x + he

_i

+ he

_j

) − f (x + he

_i

) − f (x + he

_j

) + f (x)

h

²

+ O(h),

i, j = 1, ..., n, for an approximation of the Hessian matrix.

Exercise 15 (4 Points)

Let f ∈ C

¹

( R

ⁿ

, R ) be a quadratic function of the form f(x) = 1

2 hx, Qxi + hc, xi + γ,

Q ∈ R

^n×n

symmetric and positive definite, c ∈ R

ⁿ

and γ ∈ R . Let x

₀

∈ R

ⁿ

and H be a symmetric, positive definite matrix.

Define f(x) := ˜ f(H

⁻¹²

x) and x ˜

₀

= H

¹²

x

₀

. Let (˜ x

_k

)

k∈N

be a sequence generated by the steepest descent method,

˜

x

_k+1

= ˜ x

_k

+ ˜ t

_k

d ˜

_k

with d ˜

_k

= −∇ f(˜ ˜ x

_k

)

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and t ˜

_k

= t( ˜ d

_k

) the optimal stepsize choice as determined in Exercise 5.

Let (x

_k

)

k∈N

be generated by the gradient-like method with preconditioner H, x

k+1

= x

k

+ t

k

d

k

with d

k

= H

⁻¹

(−∇f (x

k

))

and t

_k

= t(d

_k

) the optimal stepsize choice as determined in Exercise 5.

Show (by induction) that the two optimization methods are equivalent, i.e., for all k ∈ N it holds:

x

_k

= H

⁻¹²

x ˜

_k

.